Langevin Dynamics with Space-Time Periodic Nonequilibrium Forcing

Abstract

We present results on the ballistic and diffusive behavior of the Langevin dynamics in a periodic potential that is driven away from equilibrium by a space-time periodic driving force, extending some of the results obtained by Collet and Martinez in (J Math Biol, 56(6):765–792 2008). In the hyperbolic scaling, a nontrivial average velocity can be observed even if the external forcing vanishes in average. More surprisingly, an average velocity in the direction opposite to the forcing may develop at the linear response level—a phenomenon called negative mobility. The diffusive limit of the non-equilibrium Langevin dynamics is also studied using the general methodology of central limit theorems for additive functionals of Markov processes. To apply this methodology, which is based on the study of appropriate Poisson equations, we extend recent results on pointwise estimates of the resolvent of the generator associated with the Langevin dynamics. Our theoretical results are illustrated by numerical simulations of a two-dimensional system.

Notes

Acknowledgments

This work was initiated while GP was visiting the INRIA team MICMAC (now MATHERIALS) at CERMICS. The hospitality and financial support from INRIA are greatly acknowledged. RJ’s research is supported by the EPSRC through grant EP/J009636/1. GP’s research is partially supported by the EPSRC through grants EP/J009636/1 and EP/H034587/1. GS’s research is partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492. The authors benefited from discussions with Stefano Olla and Stephan De Bièvre.